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It is critical to know at what prestige level your Pokémon will be kicked-out of a gym so that you could launch a counter-offensive to increase prestige. How do you calculate this number?

For sure, adding an extra guy requires 10,000 prestige. But by that math, my gym should have been 90,000 prestige.

For example, I was in a level 9, (43202 / 50000) prestige gym and 3rd from the bottom. At what prestige level did my guy get dropped?

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Since a gym holds one Pokémon per level the maths are pretty simple:

Count the Pokémon that are ranked higher. At exactly this gym level you're out. Below you'll find the prestige required to hold or get that level.
So, if you sit on a level 9 gym and you're on 7th position you'll get kicked after the gym dropping to level 6, because 6 people are higher than you and the gym can only hold 6 Pokémon on that level.

If you want to count the Pokémon below yourself you can just count the level losses. Each loss = one down. Same example again: 3 losses = you're out. Level 9 - 3 = 6 ...after reaching level 6 you're gone.

To maintain at least level 7 to stay in the gym you need 20,000 prestige. Of course that counts for both ways.

Level | Prestige
---   | ---
1     | 0
2     | 2000
3     | 4000
4     | 8000
5     | 12000
6     | 16000
7     | 20000
8     | 30000
9     | 40000
10    | 50000
0

In a level 9 gym the next Pokemon to be kicked out will be the first in line and this will happen at 40000 prestige. Then again at 30000 and 20000. Afterwards it will decrease with increments of 4000 (16000, 12000, 8000, 4000) and then 2000 (2000, 0). So your Pokemon, being third in the gym will be kicked out if the prestige drops below 20000.

You can find more informaiton here.

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